Eckel, Carsten und Irlacher, Michael:
Multi-Product Offshoring
Munich Discussion Paper No. 2014-29
Department of Economics
University of Munich
Volkswirtschaftliche Fakultät
Ludwig-Maximilians-Universität München
Multi-Product O¤shoring
Carsten Eckel
University of Munich,
CESifo, and CEPR
yMichael Irlacher
University of Munich
zJune 25, 2014
Abstract
In this paper, we incorporate o¤shoring of labor-intensive goods in a model with multi-product …rms, and explore its implications in partial and general oligopolistic equilibrium. We identify important aspects of this phenomenon and argue that im-provements in o¤shoring opportunities can a¤ect the geographic organization of a …rm and its product range. Multi-product …rms internalize supply linkages (‡exible man-ufacturing) and demand linkages (cannibalization e¤ect). In partial equilibrium, we …nd that more products are produced o¤shore on a larger scale and …rms expand their product range with better prospects for o¤shoring. We identify the cannibalization e¤ect as an important transmission mechanism within multi-product …rms and show that the latter e¤ect hits domestic labor demand in addition to the well-known relo-cation e¤ect. Interestingly in general equilibrium these e¤ects lead to adjustments in domestic factor prices and may cause a partial re-relocation of product lines.
Keywords: Multi-Product Firms, Cannibalization E¤ect, Product Range, E¢ciency-seeking O¤shoring, General Oligopolistic Equilibrium.
JEL Classi…cation: F12, F 23, L23
We thank Daniel Baumgarten, Lisandra Flach, J. Peter Neary, Tobias Seidel, Mirjam Wuggenig, Stephen Yeaple, and participants in International Economics Workshop in Göttingen 2012, ETSG 2012 in Leuven, O¤shoring and International Production Conference in Tuebingen 2013, Midwest Economic Theory and In-ternational Trade Meetings in Ann Arbor 2013, EGIT conference at CESifo 2014, and SFB TR 15 conference in Mannheim 2014.
yDepartment of Economics, D-80539 Muenchen, Germany; tel.: (+49) 2180 - 5824 ; e-mail:
[email protected]; internet: http://www.intecon.vwl.uni-muenchen.de/.
zCorresponding author. Department of Economics, D80539 Muenchen, Germany; tel.: (+49) 2180
1
Introduction
In the last decades, progress in communication and information technologies has changed the international organization of production. Markets are dominated by large multinational …rms that control and manage production lines on a global scale. Global production networks enable …rms to bene…t from the generally lower labor costs in emerging countries. Against this background, industrialized countries fear a decline of jobs and pressure on wages. Recent academic research has identi…ed two main channels by which o¤shoring a¤ects domestic labor demand. Firstly, there is a relocation e¤ect from the displacement of tasks that formerly were carried out domestically. Secondly, there are e¢ciency gains from vertical specialization that bene…t domestic workers and increase domestic labor demand.1
In this paper, we study the consequences of a di¤erent kind of o¤shoring: O¤shoring of production lines within multi-product …rms (MPFs). Analyzing …rms that o¤er a bundle of horizontally linked products leads to important new insights into the e¤ects of o¤shoring. Our results are crucially di¤erent from the well-known e¤ects of relocating just parts of a production process of a single product. In particular, we argue that the e¢ciency e¤ect of o¤shoring can only occur if the production of a single good is linked sequentially in two or more countries. Moreover, at least part of the production stages have to remain in the home country so that domestic employment can bene…t from the higher productivity. If, however, the total production line is relocated, the latter e¤ect vanishes. A …rm which produces a range of products can decide for each product where it is produced most e¢ciently. We will show that it is the labor intensity of each product which determines its optimal production location.
The relocation of a complete production process not only prevents the e¢ciency e¤ect of o¤shoring but causes a cannibalization e¤ect of o¤shoring that has not been discussed in the literature. If the products within the product range of an MPF are horizontally di¤erentiated, the introduction of a new product will create a negative demand externality on all other products of this …rm. This is typically referred to as the cannibalization e¤ect and plays a big role in our analysis. Giving a …rm the opportunity to o¤shore production will lead to a relocation of labor-intensive products and to an extension of the product range with additional products. We show that both operations will cannibalize output of domestically produced goods and reduce demand for labor in the home country.
A growing literature on MPFs stresses horizontal relationships between products within the boundaries of a single …rm and analyzes the e¤ects of globalization on the product range
1The e¢ciency or productivity e¤ect of o¤shoring is stressed in recent contributions to the o¤shoring
literature, such as Eckel (2003), Grossman and Rossi-Hansberg (2008), Rodriguez-Clare (2010), and Egger et al. (2013).
of a …rm. Bernard et al. (2010) emphasize this as a new margin of …rm adjustment, which Eckel and Neary (2010) refer to as intra-…rm extensive margin. Within a multi-product framework, we investigate how improvements in the opportunities for o¤shoring a¤ect the geographic organization and the product range of an MPF. For this purpose, we set up a general oligopolistic equilibrium (GOLE) model with MPFs and enrich this framework by introducing the …rm’s opportunity to o¤shore the production of multiple varieties to a low-wage emerging country. Varieties within a …rm’s product line are linked on the cost side through a ‡exible manufacturing technology, which captures the idea that - besides a core competence - an MPF can expand its portfolio with varieties that are less e¢cient in production. When producing abroad, a …rm can use the same production technology as in the home country but additionally it has to bear o¤shoring costs.
We derive our results in partial and in general equilibrium. As a main result in partial equilibrium, we …nd that more products are produced abroad when prospects for o¤shoring improve. Furthermore, savings from lower o¤shoring costs lead to an extension of the prod-uct portfolio as the opportunity to produce labor-intensive prodprod-ucts abroad enlarges the pro…t maximizing product range of an MPF. In a model where …rms internalize demand linkages, rising outputs of foreign-produced varieties and additional varieties in the portfolio are crowding out domestic production, that does not bene…t from lower o¤shoring costs. We stress this cannibalization e¤ect as an important transmission channel that is speci…c to MPFs. In our model, in addition to the well-established relocation e¤ect, cannibalization hits domestic production. For this reason, the analysis in partial equilibrium clearly indicates that domestic labor demand will decrease in the presence of more o¤shoring.
In general equilibrium, our analysis highlights adjustments through factor markets as an important transmission channel of external shocks on both the cuto¤ variety and the product range.2 With endogenous domestic wages the results are not as clear cut anymore.
It is no longer apparent that more products will be produced o¤shore with falling o¤shoring costs. We show that the more domestic production bene…ts from falling domestic wages the more likely is the partial result reversed in general equilibrium. Therefore, our model is able to predict patterns in which …rms "re-relocate" entire product lines following a decline in o¤shoring costs and a delayed fall in wages.
Our paper builds on and extends two strands of the existing literature in international trade on both MPFs and o¤shoring. Particularly with regard to the connection of both strands, Baldwin and Ottaviano (2001) come up with a multi-product setting where oligopolis-tic …rms may produce some varieties in one country and other varieties in another. However,
2The cuto¤ variety is de…ned as the product where the …rm is indi¤erent concerning the optimal
they explain intra-…rm trade patterns akin to reciprocal dumping à la Brander and Krug-man (1983) and not via factor price di¤erences across countries. Hence, their approach is not associated with “o¤shoring” per se. In a recent paper, Yeaple (2012) extends a framework by Bernard et al. (2011) with a proximity-concentration tradeo¤. In his setting, …rms pro-duce multiple products for multiple countries and choose whether to export from the home country or to manufacture locally. Unlike to our paper, his focus is not on wage di¤erentials between countries but on …rm heterogeneity with respect to managerial expertise. Managers deliver expertise to foreign a¢liates, which means that …rms with a higher manager e¢ciency tend to build foreign a¢liates rather than to export to foreign countries. In an empirical analysis, McCalman and Spearot (2013) examine the role of vertical product di¤erentiation in the decision where to produce a speci…c variety. Using a dataset of light truck sales in the US, Canada and Mexico, they study the location decision of …nal assembly. The pat-terns of o¤shoring that they …nd can be explained by the labor intensity in the automobile production. Furthermore, it is consistent with one of our theoretical predictions that foreign output is produced at a lower scale.
We also contribute to the large literature on international production. Our way to de-termine the cuto¤ variety between domestic and foreign production is reminiscent of a key contribution to the o¤shoring literature by Feenstra and Hanson (1996). While in their theoretical model, o¤shoring takes the form of relocating labor-intensive activities of a sin-gle manufactured good, they adopt a more general de…nition of o¤shoring in the empirical part.3 Next to imports of intermediate goods, they further include …nal goods that are sold
under the brandname of a …rm in their de…nition of o¤shoring. Therefore, this measure-ment of o¤shoring is directly related to our way of de…ning o¤shoring as the relocation of complete production lines. By including …nal goods in their measure of o¤shoring, Feenstra and Hanson do better at explaining wage patterns and employment changes for the United States.4 Other empirical papers measure o¤shore activity by the total employment of foreign
a¢liates. Using this kind of measure, authors typically think of capturing the relocation of vertically related tasks or the replication of domestic production abroad (horizontal FDI). However, we show that this way of measuring international activity is perfectly consistent to what we call multi-product o¤shoring.
Existing theoretical research on MPFs is concerned mainly with the product market side of the economy. The main question which is tried to be answered is how MPFs absorb inter-national trade. Intra-…rm product switching is frequent and contributes like …rm entry and
3Feenstra and Hanson (1996) refer to this phenomenon as outsourcing.
4Feenstra and Hanson (1996) argue that previous studies like Berman et al. (1994) and Lawrence (1994)
exit to the evolution of aggregate outcomes in an industry.5 The literature di¤ers in the way
of modelling the demand for and the decision to supply multiple products and in the assump-tions about market structure. Most recent models assume that markets can be characterized by monopolistic competition, in which …rms produce a large number of products but are themselves in…nitesimal small in scale in the economy (see Arkolakis and Muendler (2010), Bernard et al. (2011), Nocke and Yeaple (2013), and Mayer et al. (2014)). Our model is built along the lines of Eckel and Neary (2010) who set up a di¤erent approach and assume that markets are oligopolistic. Their underlying market structure highlights as an important feature the cannibalization e¤ect, which also plays a crucial role in our model.6 Next to these
demand linkages, Eckel and Neary’s approach incorporates cost linkages between varieties in the form of ‡exible manufacturing.7
The remainder of the article is structured as follows. Section 2 recaps the basic model of Eckel and Neary (2010) and incorporates o¤shoring into this framework. Subsequently, we provide comparative static results of falling o¤shoring costs. Section 3 shows how these results transform when wages are endogenized in general equilibrium. Section 4 concludes and summarizes results. Mathematical derivations and a numerical simulation of our model are presented in the Appendix.
2
The Model
To conduct our analysis, we rely on the multi-product framework with ‡exible manufacturing proposed by Eckel and Neary (2010). We introduce a model where …rms on grounds of e¢ciency seeking can relocate the production of labor-intensive goods abroad. Our setup consists of two countries, Home and Foreign, and a large world market. There is a continuum of identical industries in Home, whereby the output produced in each of these industries is sold on the world market. Foreign is a low wage emerging country and acts as a potential destination for an a¢liate. We begin this section with the analysis of one single sector by considering the behavior of the consumers in the world market and the optimal …rm behavior in this industry.
5Bernard et al. (2010) report changing product ranges for more than 50 percent of US …rms within …ve
years whereby one-half of those …rm both added and dropped at least one product.
6The cannibalization e¤ect is also considered in recent articles by Feenstra and Ma (2008) and Dhingra
(2013).
7The concept of ‡exible manufacturing is also used in Milgrom and Roberts (1990), Eaton and Schmitt
2.1
Consumer Behavior: Preferences and Consumer Demand
We assume that LW consumers in the world market maximize their utility de…ned over theconsumption of di¤erentiated products. Referring to the model of Eckel and Neary (2010), we maintain the speci…cation of preferences in the form a two-tier utility function.8 The
upper tier is an additive function of a continuum of sub-utility functions over industries z, where z varies over the interval [0; 1], given by
U [u (z)] = Z 1
0
u (z) dz. (1)
The representative consumer’s sub-utility is de…ned over per variety consumption q(i; z) with i 2 and total consumption Q Ri2 q(i; z)di, where is a set of di¤erentiated goods o¤ered in industry z. To be more speci…c, we assume
u (z) = aQ 1
2b (1 e)
Z
i2
q(i; z)2di + eQ2 . (2)
Eq. (2) has a standard quadratic form, where a, b denote non-negative preference parameters and e is an inverse measure of product di¤erentiation which lies between 0 and 1. Lower values of e imply that products are more di¤erentiated and hence less substitutable. In the event of e = 1, consumers have no taste for diversity in products and demand depends on aggregate output only. Consumers maximize utility in Eqs. (1) and (2) subject to the budget constraint R01Ri2 p(i; z)q(i; z)didz I, where p (i; z) denotes the price for variety i in industry z and I is individual income. This yields the following linear inverse individual demand function:
p(i; z) = a b [(1 e)q(i; z) + eQ] , (3)
where is the marginal utility of income, the Lagrange multiplier attached to the budget constraint. Market-clearing imposes that each …rm faces a market demand x(i; z) that con-sists of the aggregated demand of all consumers in the world market LWq(i; z). For the
inverse world market demand, we get p(i; z) = a0
b0
[(1 e)x(i; z) + eX] , (4)
where a0 a is the consumers’ maximum willingness to pay and b0 b
LW is an inverse
measure for the market size. Finally, X R0 x (i; z) di represents the total volume of varieties produced and consumed in industry z. Note that X is de…ned over the goods actually
8These preferences combine the continuum quadratic approach to symmetric horizontal product
consumed with i 2 [0; ], which is a subset of the potential products . With no quasi-linear term in Eq. (2), the value of is not constant, which implies that a0
and b0
are endogenously determined in general equilibrium. However, with a continuum of industries, we may assume that each …rm takes these parameters as given. Hence, each …rm has market power in its own market but it is small in the economy as a whole. This assumption permits a consistent analysis of oligopoly in general equilibrium. As it has become standard in the literature, we choose the marginal utility of income as the numeraire and set equal to one (see Neary (2009) for further discussion).
2.2
Firm Behavior: Costs and Technology of MPFs
This section considers technology and optimal …rm behavior in industry z.9 We focus on
intra-…rm adjustments, so competition between …rms plays only a second-order role. To keep the analysis as simple as possible, we focus on the monopoly case. Extending the analysis to oligopoly is straightforward.10 According to that, each industry z is characterized by exactly
one …rm whose objective it is to maximize pro…ts by choosing both the scale and scope of production, as well as choosing the optimal location for producing each speci…c variety. When choosing the optimal location for production, …rms seek to reduce costs by producing labor-intensive goods o¤shore where a comparative advantage exists due to lower wages. For simplicity, we assume no …xed costs for both domestic and foreign production.
In our model, an MPF is characterized by a core competence and ‡exible manufacturing. Technology is …rm-speci…c and, therefore, it can be applied correspondingly in Home and in Foreign. As in Grossman and Rossi-Hansberg (2008), technology is transferable as a home …rm will use its own technology when performing a task abroad. Flexible manufacturing is characterized by one core competence, in which the …rm is most e¢cient in fabrication. Furthermore, an MPF can produce additional varieties with rising marginal costs.
Production costs in our model comprise both a product-speci…c and a monitoring compo-nent (managerial e¤ort) which we assume to be zero for production at home. This assumption implies that the ability to monitor varies with distance. Managerial e¤ort is needed to su-pervise production and to provide the …rm’s technology abroad.11 By incorporating these
costs, we try to capture the more general idea that aggravated monitoring through
man-9We concentrate on symmetric industries and drop the industry index z in the following analysis. We
consider this index again when we aggregate over all industries and turn to the level of the economy as a whole in general equilibrium.
10The interested reader is referred to the Appendix in Eckel et al. (2011).
11See for example Grossman and Helpman (2004). They assume that a principal is able to observe the
manager’s e¤orts at a lower cost when the manager’s division is located near to the …rm’s headquarters as compared with when it is located across national borders.
agers, less skilled workers, worse infrastructure, or inferior contractual enforcement, a¤ect production in emerging countries. In the following analysis, we refer to this cost component as o¤shoring costs. To put it formally, we assume a Ricardian technology where domestic (foreign) production costs c(i) (c (i)) are given by
c(i) = w (i) and (5)
c (i) = w ( (i) + t), (6)
with (i) denoting the labor input coe¢cient for variety i, w (w ) being the wage level in Home (Foreign) and …nally t representing the o¤shoring costs.12 Latter is measured in labor
costs and is the same for all products assembled abroad. As we are analyzing the relocation of total production lines and not the relocation of just parts of a production process, the assumption that t is identical for all o¤shored varieties seems fair. Technology is …rm- and not country-speci…c, therefore (i) is the same in both countries. We assume the following
properties: (0) = 0 and @c
@i = @
@iw > 0.
Closed Economy: Without o¤shoring, optimal …rm behavior is composed of
maximiz-ing total …rm pro…ts both with regard to scale and to scope. Considermaximiz-ing the technology assumptions above and denoting the scope of the product portfolio by , pro…ts are given by
= Z
0
[p(i) c(i)] x(i)di. (7)
Firms simultaneously choose the quantity produced of each good and the mass of products produced. Maximizing pro…ts in Eq. (7) with respect to scale x (i) implies the …rst-order condition for scale:
@
@x(i) = p(i) c(i) b
0
[(1 e)x(i) + eX] = 0 (8)
that leads to the optimal output of a single variety x(i) = a
0
w (i) 2b0
eX
2b0(1 e) (9)
with X R0 x(i)di denoting total …rm scale.13 The negative impact of total …rm scale X
on the output of a single variety displays the cannibalization e¤ect: @x(i)@X = e
(1 e) < 0.
An MPF internalizes the e¤ect that increasing output of a certain variety lowers prices for this, as well as, for all other varieties in the …rm’s product range. This e¤ect only exists if e > 0, i.e. if products are not perfectly di¤erentiated. Furthermore, Eq. (9) shows that, given its total output, a …rm produces less of each variety the further away it is from its core competence. Given the symmetric structure of demand, this means that a …rm charges higher prices for products that are further away from its core competence (see Eckel and Neary (2010), p.193 for a detailed analysis).
In the next step, we consider the …rm’s optimal choice of product line. MPF’s add new products as long as marginal pro…ts are positive. Maximizing Eq. (7) with respect to scope implies the respective …rst-order condition14
@
@ = [p( ) c( )]x( ) = 0. (10)
From Eq. (8), we know that the pro…t on the marginal variety [p( ) c( )] cannot be
zero. The …rm adds new varieties up to the point where the marginal cost of producing the marginal variety equals the marginal revenue at zero output. The pro…t maximizing product range implies that the output of the marginal variety x( ) is zero. Using Eq. (9) and setting x ( ) equal to zero yields
c( ) = a0
2b0
eX. (11)
Comparing Eqs. (9) and (11), we see that …rms add new varieties to their product portfolio until optimal output of the marginal variety falls to zero. Inspecting Eq. (11) reveals the cannibalization e¤ect which in‡uences the scope of production: @X@ = @c( )=@b0e < 0. Figure 1 illustrates the …rst-order condition for scope and determines the pro…t-maximizing product range.
[Insert Figure 1 about here]
Open Economy: So far, we have implicitly assumed that the o¤shoring costs t were
prohibitively high, so that all production was located in the home country. As globalization
13The second-order condition of this maximization problem is: @2
@x(i)2 =
@p(i)
@x(i) b0(1 e) b0e @X @x(i)<0. 14The second-order condition of this maximization problem is: @2
@ 2 = [p ( ) c( )] @x( ) @ <0, as @c( ) @ >0 and, thus, @x( )@ = 1 2b0(1 e) @c( ) @ <0.
leads to improvements in information technology and reductions in communication costs, we analyze a decrease in the parameter t, which implies that …rms can enjoy bene…ts of lower factor prices and thus gains from relocating labor intensive products to a low-wage location. In our model, the motive for o¤shoring is e¢ciency-seeking, which means that the necessary condition for o¤shoring is: w < w. The su¢cient condition for o¤shoring is that the o¤shoring costs are below a critical value: t < tcrit. The critical value of o¤shoring costs
can be calculated as
tcrit = (a
0
2beX)(w w )
ww . (12)
It is straightforward to see that the critical value of o¤shoring costs is rising in the wage di¤erential between Home and Foreign.
In the analysis below, we refer to cases in which o¤shoring cost are su¢ciently low, so there is a fragmentation of production into domestic and foreign-produced varieties. We de…ne e as the cuto¤ variety. For variety e, the …rm is indi¤erent concerning its optimal pro-duction location. Varieties with a lower labor input coe¢cient than e are produced onshore, whereas varieties with a higher labor input coe¢cient are produced o¤shore. Combining Eqs. (6) and (9), gives the optimal scale of a foreign-produced variety:
x (i) = a
0
w ( (i) + t) 2b0
eX
2b0(1 e) . (13)
Given that the marginal variety is produced in Foreign, the pro…t maximizing product range is de…ned by
w ( ( ) + t) = a0
2b0
eX. (14)
In the open economy, an MPF faces a third maximization problem, next to optimal scale and scope of production. Now, the …rm has also to determine the pro…t maximizing geographic location of production. Analogous to Eq. (7), total pro…ts in the open economy are given by
=
e
Z
0
(p(i) c(i)) x(i)di + Z
e
(p (i) c (i)) x (i)di, (15)
with the …rst integral being total pro…ts from domestic production and the second integral being the equivalent for foreign production. With Eq. (8) and total …rm output X being composed of domestically and foreign-produced goods as
X = e Z 0 x(i)di + Z e x (i)di, (16)
we can rearrange Eq. (15): = (1 e)b0 2 6 4 e Z 0 x(i)2di + Z e x (i)2di 3 7 5 + b0eX2. (17)
Maximizing Eq. (17) with respect to the optimal cuto¤ of production e leads to
x(e) = x (e). (18)
Formal details of the derivation can be found in the Appendix.
Lemma 1 An MPF chooses the optimal cuto¤ level of production e exactly at that product where optimal scale in Home and in Foreign are the same. Combining Eqs. (9) and (13), this means that for variety e the …rm is just indi¤erent concerning the location of production because costs are identical, i.e.
w (e) = w ( (e) + t). (19)
To visualize our analysis, we illustrate the e¤ects of falling o¤shoring costs in Figure 2. In Figure 2a), production of the whole portfolio is accomplished in Home as o¤shoring costs are prohibitively high. In Figure 2b), o¤shoring cost are below the critical value in Eq. (12). We observe that varieties i 2 [0; e] are still produced in Home, as their production is e¢cient enough, so the bene…ts of lower foreign wages do not prevail the o¤shoring costs. Production of varieties i 2]e; old[ is relocated, as these goods can be produced at a lower cost in Foreign. Products i 2] old; [ constitute an extension of the …rm’s product range. The MPF adds these varieties at the intra-…rm extensive margin, whereby these goods would not be o¤ered in case of producing exclusively in Home. The speci…cation of our model suggests that an MPF produces exactly those varieties o¤shore, where its e¢ciency is relatively low.
[Insert Figure 2 about here]
We conclude this section with a graphical illustration of the main properties of our model in Figure 3. The graph portrays optimal scale of production for the entire portfolio across the two production locations. We will use this graph in the next section as a useful tool in the comparative statics analysis. Figure 3 shows that due to the underlying ‡exible manufacturing technology, output of the core competence is the highest. At the cuto¤ e
x e = x e the …rm switches to foreign production. Therefore, the slope of the curve changes at this point. Finally, the pro…t maximizing product range is pinned down at x ( ) = 0.
[Insert Figure 3 about here]
2.3
Comparative Statics
We still assume that t is below its critical value determined in Eq. (12), so the …rm engages in foreign production. In the comparative statics, we analyze the e¤ect of better prospects for o¤shoring on the geographic organization (optimal cuto¤) and on the pro…t-maximizing product range. Furthermore, we investigate the impact of reduced costs of o¤shoring on the output of domestic and foreign-produced varieties, as well as on total …rm output. These endogenous variables of our model x(i), x (i), , X, and, e are determined in Eqs. (9), (13), (14), (16), and, (19) respectively. Totally di¤erentiating this system of equations generates the comparative-static e¤ects of decreasing o¤shoring costs t.
Recent academic research on MPFs brings forth varying results on the e¤ects of global-ization on the product range of a …rm. A set of papers, including Eckel and Neary (2010), Bernard et al. (2011), and Mayer et al. (2014) show that MPFs will reduce their product ranges in response to trade liberalization. Increased competition forces …rms to drop their worst performing products. In Feenstra and Ma (2008), increasing the market size leads to an expansion of the product range. Very recently, Qiu and Zhou (2013) show that the most productive …rms in an economy may expand their product scope after globalization. In this paper, we do not focus on the competition and market size e¤ects of globalization. Glob-alization does also mean that access to foreign production locations is facilitated. Having the latter interpretation in mind, we can clearly show that the product scope increases in response to globalization.
Proposition 1 If t is below the critical value determined in Eq. (12), falling o¤shoring costs induce an MPF to add new products at the intra-…rm extensive margin, i.e.
d ln d ln t = 2t 1 0( ) < 0, (20) where: 1 = (1 e + e ) > 0 and 2 = 1 e + e~ :
This result can be visualized in Figure 2b). A decrease in t corresponds to a downward shift of the c -curve which indicates an extension of the product range.
In a next step, we want to discuss the e¤ects of globalization on the domestic product range ~. With respect to the large literature on international production, this aspect has been neglected so far in theoretical models. We …nd that better prospects for o¤shoring reduce the domestic product range and incentivize a …rm to relocate marginal varieties. Proposition 2 Falling o¤shoring costs make foreign production more attractive and thus lead to an e¢ciency-seeking relocation of production from the high-wage country to the low-wage country, i.e.
d ln ~ d ln t =
w t
(w w ) 0 ~ ~
> 0. (21)
As the wage rate in the home country w is higher than abroad w , the expression is strictly positive. The magnitude of this e¤ect can be shown to depend on the point elasticity
of the cost curve at the marginal variety: (~) 0 ~ ~= ~ . The latter stands for
an inverse measure of ‡exibility of an MPF. High values of (~) imply that a change in
~ will cause a large change in marginal costs. Hence, the change in the domestic product range following globalization will be smaller, the stronger domestic production costs react to a marginal decrease in ~. To see this, we can rewrite Eq. (21) in d ln ~=d ln t = 1= (~)
using the indi¤erence condition in Eq. (19). In Figure 2b), a decrease in t corresponds to a downward shift of the c (i) -curve which is equivalent to shifting production abroad (e falls). Former domestically produced goods are now produced abroad. Referring to previous discussion, the e¤ect is less pronounced in the case of steep cost curves.
So far, we have analyzed within-…rm adjustments at the intra-…rm extensive margin. In the next step, we focus on the output pro…les (intensive margin) of domestically and foreign-produced varieties. Following a fall in t, o¤shore production gets cheaper and, therefore, foreign varieties are produced at a larger scale.
Proposition 3 If t is below the critical value determined in Eq. (12), falling o¤shoring costs induce the …rm to increase outputs of all foreign-produced varieties, i.e.
d ln x (i) d ln t = w t 2b0(1 e) x (i) 2 1 < 0. (22)
As an important feature in our model, we emphasize demand linkages between varieties in the product portfolio of a …rm. Falling o¤shoring costs do not reduce domestic production costs but indirectly a¤ect domestic output through the cannibalization e¤ect. Rising output of foreign production crowds out domestic production as domestic varieties internalize the cannibalization e¤ect.
Proposition 4 The cannibalization e¤ect induces an MPF to reduce outputs of all domes-tically produced varieties in consequence of falling o¤shoring costs, i.e.
d ln x (i) d ln t = e ~ 1 w t 2b0(1 e) x (i) > 0. (23)
In the case of perfectly di¤erentiated varieties, i.e. e = 0, domestic output is independent of foreign production and hence, the derivative in Eq. (23) is zero. With e being positive, varieties become substitutable and domestic output is crowded out by foreign production. However, it is straightforward to show that despite lower domestic output, total …rm output X is increasing with falling o¤shoring cost. The positive impact of rising foreign output combined with the extension of the product range outweighs the negative impact of falling domestic output on total …rm scale.
Proposition 5 With falling o¤shoring costs, an MPF increases total …rm output because of the higher scale of foreign-produced varieties and the extension of the product portfolio, i.e.
d ln X d ln t = w ~ t 2b0 1X < 0. (24)
Formal details of all the derivations can be found in the Appendix. To illustrate the e¤ects of falling o¤shoring costs, we draw on the graphical tool developed in Figure 3. In Figure 4, the dotted line represents the situation after the reduction in t. Inspecting this graph reveals two negative e¤ects on domestic production: A relocation e¤ect from shifting production abroad and a cannibalization e¤ect from rising foreign output. The latter e¤ect is a new transmission channel speci…c to MPFs that we want to highlight. It results from the fact that with lower production costs abroad, output of foreign varieties and the foreign product range will increase. These intra-…rm adjustments crowd out the production of domestic varieties which does not bene…t from lower production costs abroad. The main comparative static results are indicated by the arrows in Figure 4.
[Insert Figure 4 about here]
2.4
Implications for the Measurement of O¤shoring
From a theoretical point of view, the way we are thinking about o¤shoring as a relocation of production lines within MPFs is novel. However, the manner how o¤shoring is measured in
the broad empirical literature on international production is similar to our de…nition. The measure of outsourcing which is used in Feenstra and Hanson (1996) is directly related to our de…nition, as it includes also …nal goods next to imported intermediates. The authors argue that this "must be included in any valid measure of outsourcing" (Feenstra and Hanson (1996), p.107). Many other papers that discuss o¤shoring from an empirical perspective use measurements of o¤shoring that respond not only to a relocation of vertically related processes, but also respond to what we call multi-product o¤shoring. Papers such as Head and Ries (2002), Ebenstein et al. (2012), and Becker et al. (2013) measure o¤shoring activity in an industry by the total employment of foreign a¢liates. Using employment in foreign a¢liates as a measure for o¤shoring is perfectly in-line with our model. To underline that measuring o¤shoring like this could also mean the type of o¤shoring that we have in mind, we calculate the total employment in foreign a¢liates and show how it responds to better o¤shoring opportunities. In industry z, labor demand l for foreign-produced varieties is given by l (z) = (z) Z e(z) (i)x (i)di. (25)
It is determined by the scale and scope of foreign-produced varieties i 2 [e; ]. We derive total labor demand in the o¤shore destination L by integrating over all industries z 2 (0; 1)
L = 1 Z 0 l (z) dz = 1 Z 0 (z) Z e(z)
(i; z)x (i; z)didz. (26)
By substituting for x (i) and evaluating the integral, we come up with the following equation
L = ~ (a0 2b0 eX w t) 0 w 00 2b0 (1 e) ; (27) where 0 1 ~ R
~ (i) di is the mean labor input of foreign-produced varieties and 00 1
~
R
~ (i)2di is the second moment around zero of the distribution of labor requirements.
We totally di¤erentiate Eq. (27) and analyze again the e¤ects of better prospects for o¤-shoring: d ln L d ln t = w t 2b0(1 e) L 8 < : ~ 0 2 1 + ~w ( ) ~ (w w ) (~) 9 = ;< 0. (28)
The latter expression clearly indicates that the total employment of foreign a¢liates is in-creasing in falling o¤shoring costs. Therefore, measuring o¤shore activity by total employ-ment of foreign a¢liates captures the type of o¤shoring that we have in mind.
Lemma 2 Falling o¤shoring costs increase total employment in the o¤shoring destination.
3
General Equilibrium
The previous section analyzed the e¤ects of falling o¤shoring costs on the product range, per variety output, total …rm output and the optimal location of production. Up to this point, the approach was partial, since we did not consider endogenous changes in wages. Our analysis in partial equilibrium clearly yields a fall in domestic production, because, on the one hand, per variety output of domestic varieties gets crowded out and, on the other hand, varieties close to the cuto¤ are relocated with falling o¤shoring costs. In the next steps, we focus on new insights into the labor market e¤ects from o¤shoring which arise from the framework that we have presented so far. For this purpose, we introduce a simple labor market and show how domestic labor demand is a¤ected by multi-product o¤shoring. Subsequently, we analyze again the comparative statics exercise of falling o¤shoring costs under consideration of labor market clearing.
3.1
Labor Market Clearing
In this section, we turn to the level of the economy as a whole and explore the general equilibrium e¤ects of falling o¤shoring costs. To simplify the analysis, we assume that all industries are identical. In a …rst step, we need to specify how wages are determined. We assume a total labor supply LS, that is supplied inelastically by the households in Home.
Domestic labor demand in industry z is given by
l (z) = Z e(z)
0
(i) x (i) di. (29)
It is determined by the scale and scope of domestically produced varieties i 2 [0; e]. We derive total labor demand L in our economy by integrating over all industries z 2 (0; 1):
L = 1 Z 0 l (z) dz = 1 Z 0 e(z) Z 0
Our main interest in this section is to determine the labor market e¤ects of o¤shoring. In the previous section we have identi…ed two e¤ects of falling o¤shoring costs: A relocation e¤ect and a cannibalization e¤ect. The relocation e¤ect a¤ects the marginal variety e and the cannibalization e¤ect a¤ects the scale of domestic production x (i). Totally di¤erentiating domestic labor demand in Eq. (30) with respect to t yields:
d ln L d ln t = e L 8 > < > : e x e d ln e d ln t | {z } relocation e¤ect + 0 x (i)d ln x (i) d ln t | {z } cannibalization e¤ect 9 > = > ;> 0, (31) with 0 1 e Re
0 (i)di being the mean labor input of domestically produced varieties.
15 The
…rst part of Eq. (31) describes the relocation e¤ect and the second part stands for the canni-balization e¤ect. Latter e¤ect is new and is speci…c to MPFs. With falling o¤shoring costs, scale of foreign production rises because of lower production costs abroad. This behavior cannibalizes domestic production and reduces domestic labor demand.
Lemma 3 For a given domestic wage, falling o¤shoring costs reduce domestic demand for
labor through two channels. A relocation e¤ect leads to a shift of labor-intensive domestic products abroad. Furthermore, domestic production internalizes a cannibalization e¤ect of rising foreign output and is crowded out.
In equilibrium, wages must adjust to ensure that total labor supply LS equals total labor
demand determined by the cuto¤ of domestic production e in all industries z 2 (0; 1). This is re‡ected by the following labor-market equilibrium condition for the home country:
LS = 1 Z 0 l (z) dz = 1 Z 0 e(z) Z 0
(i; z)x(i; z)didz. (32)
We can now substitute for x(i) and evaluate the integral to obtain
LS = e (a 0 2b0 eX) 0 w 00 2b0(1 e) , (33) where 00 1 e Re 0 (i)
2di stands for the second moment around zero of the distribution of
labor requirements. Combining Eq. (33) with the system of equations from the analysis in partial equilibrium, we can use the respective equations for investigating how …rm-level
15From inspection of propositions 2 and 4, we know that: dln e
dln t >0 and dln x(i)
adjustments respond to declining o¤shoring costs with endogenous wages. We derive the comparative statics results by totally di¤erentiating all equations of the system. Formal details of all the derivations can be found in the Appendix.
3.2
Comparative Statics in General Equilibrium
One important issue in general equilibrium which we want to analyze in the …rst place, is the e¤ect of better prospects for o¤shoring on domestic factor prices w. In the previous sections, we have identi…ed two negative impacts of o¤shoring on domestic labor demand: The relocation and the cannibalization e¤ect. However, in equilibrium, total labor supply must equal total demand for labor. To ensure this equality, domestic wages must fall. Proposition 6 With falling o¤shoring costs, ceteris paribus, foreign production gets more attractive. To ensure labor market clearing in equilibrium, there are adjustments on the labor market in the form of falling domestic wages, i.e.16
w w t d ln w d ln t = n e (w w ) 0 ~ ~ ~ 0 + 1w h ( ) ~ i ~ o> 0. (34)
Considering these labor market adjustments reveals that in general equilibrium falling o¤shoring costs not only make foreign production cheaper but also reduce production costs in the home country. The latter has important implications on the main variables of interest in our model, which we point out in the following.
With lower production costs in both countries, it is apparent that an MPF will increase its total scale:
d ln X d ln t = w ~ t 2b0 1X w~ 0 2b0 1X d ln w d ln t < 0. (35)
The mathematical derivation and an expression where the change in wages is substituted can be found in the Appendix. Eq. (35) is the general equilibrium equivalent of Eq. (24). Comparing both equations immediately points out that due to the adjustment of factor prices (represented by the second fraction), the general equilibrium e¤ect will be of greater magnitude than the partial equilibrium e¤ect (represented by the …rst fraction). Within our framework, a larger …rm scale X enhances cannibalization between varieties. Caused by falling domestic wages, the latter e¤ect leads to a new channel that we have to consider when analyzing the repercussions of falling o¤shoring costs on the product range of a …rm.
16The term =nh(1 e) + e ~ i~ 00+ e~2 2o(w w ) 0 ~ +
1 ~ w
h
( ) ~ i ~ is the determinant of the system of equations. It is positive which ensures that the equilibrium is unique and stable.
We illustrate this channel in the following equation: d ln d ln t = 2t 1 0( ) + ew~ 0 1w 0( ) d ln w d ln t < 0, (36)
where the …rst part of Eq. (36) represents the partial e¤ect which is clearly of a negative sign. The second part of Eq. (36) is the additional channel in general equilibrium arising from the adjustment of wages. This e¤ect is positive and, therefore, works in the opposite direction as it induces the …rm to increase its total output X. Inspecting Eq. (36) reveals that the general equilibrium e¤ect is switched o¤ for e being zero. With products being perfectly di¤erentiated, there is no cannibalization of the rising total …rm output X on the marginal varieties within the product range. However, we can analytically show that the result from partial equilibrium is recon…rmed even for e > 0. Therefore, the adjustments in general equilibrium only have a dampening e¤ect on the product range which is driven by the intensity of cannibalization determined by the di¤erentiation parameter e. A proof for this result is provided in the Appendix.
Proposition 7 Falling o¤shoring costs reduce costs in both production sites and hence en-large total …rm output X to a en-larger extend compared to partial equilibrium. Latter result dampens but does not reverse the e¤ect of falling o¤shoring costs on the product range in general equilibrium. The dampening e¤ect depends on the strength of cannibalization.
In the next step, we focus our attention on the optimal geographic organization of an MPF. Regarding the optimal cuto¤ of production, we identify two opposing forces in general equilibrium following a fall in the parameter t. On the one hand, there is the direct e¤ect of lower o¤shoring costs which tends to shift production abroad (observed e¤ect in partial equilibrium, see Eq. (21)). On the other hand, we …nd decreasing domestic wages which brings forth an inventive to pull back production into the home country. The latter causes an ambiguity on the total e¤ect of falling o¤shoring costs in general equilibrium which can be seen in the following derivative:
w t d ln ~ d ln t = 1 00 eh ~ ~ + ~ 0i 0 ? 0. (37)
We now focus on this ambiguity and investigate the causes that lie behind it. To begin with, Eq. (37) is positive for e being zero. With perfectly di¤erentiated products, domestic varieties do not internalize cannibalization through rising outputs of foreign varieties (com-pare Eq. (23)). Thus, there is no reducing force on domestic labor demand via a lower scale of domestically produced varieties (the cannibalization e¤ect of o¤shoring, stressed in
Eq. (31)). Thereby, to ensure labor market clearing, domestic wages will decline less and the wage-e¤ect will not dominate the better opportunities to o¤shore. With 0 < e < 1, there is the possibility that Eq. (37) gets negative, i.e. with falling o¤shoring costs even more products are produced in Home. This happens if the general equilibrium adjustment of factor prices prevails the foreign cost reduction via lower o¤shoring costs.
To get some further intuition for this ambiguity, we investigate the e¤ect of an exogenous change in domestic wages on domestic output. Di¤erentiating optimal scale x (i) in Eq. (9) with respect to the wage rate w yields:17
d ln x (i) d ln w = w 2b0 (1 e) x (i) h ee 0 1 (i) i 1 7 0. (38)
The algebraic sign of Eq. (38) behaves ambiguously. Outputs of varieties with a labor
input coe¢cient (i) far below the average 0
may even fall with falling wages (i.e. for these varieties d ln x(i)d ln w > 0). The reason for this is that varieties which are very e¢cient in production require just sparse labor input and hence, bene…t slightly from falling wages. However, these varieties fully internalize the cannibalization e¤ect through rising outputs of labor-intensive varieties which bene…t a lot from lower factor prices.18 Latter results imply
that varieties bene…t more from falling wages the higher is their respective labor input. These insights are important features of our model which can help to explain the ambiguous e¤ects of lower o¤shoring costs on the cuto¤ variety e.
From the total derivatives of our system of equations, we obtain two equations in d ln e d ln t and d ln w d ln t, given by 19 d ln ~ d ln t = w t (w w ) 0 ~ ~ w ~ (w w ) 0 ~ ~ d ln w d ln t ? 0 (39) and d ln ~ d ln t = w t ~ e 0 1 3 ~ + w 3 ~ 00 e~ 02 1 ! d ln w d ln t ? 0. (40)
17The interested reader …nds the e¤ects of an exogenous change in domestic wages on all endogenous
variables in the Appendix.
18The condition for the output of the core competence to fall with falling wages is : (0) < ee 0
1 . The
cuto¤ variety e has the highest labor input coe¢cient (e) in the domestic product range. The output of this variety x(e) rises with falling wages: ddln x(e)ln w <0.
19
3= (a0 2b0eX) w (~) > 0. From the …rst-order condition of scope it becomes obvious that this
Eq. (39) follows immediately from the determination of the pro…t maximizing cuto¤ in Eq. (19). Eq. (40) is derived after some mathematical conversion from the labor market clearing condition in Eq. (33). Inspecting Eq. (39) reveals that the partial equilibrium result (the …rst part of the expression) is more likely to be reversed, the higher the adjustment in wages is weighted. From the analysis of Eq. (38), we know that varieties with high labor inputs will bene…t more from reductions in factor prices. This insight can be reapplied to Eq. (39), where we observe the wage e¤ect to be of greater impact, the higher is the labor input at the marginal variety ~ . By analogy, we apply this intuition to Eq. (40), where it becomes apparent that the higher is the mean labor input of domestic production 0
, the more likely the domestic wage reduction outweighs the cost advantage through lower o¤shoring costs. We summarize these insights in the following proposition.
Proposition 8 In partial equilibrium, lower o¤shoring costs t lead to a distinct fall in e (i.e.
d ln e
d ln t > 0). This result does not necessarily hold in general equilibrium which implies that it is
possible that even more products are produced onshore with better opportunities of o¤shoring (i.e. d ln e
d ln t < 0). This ambiguity is caused by the general equilibrium result of falling domestic
wages. We show that the result in partial equilibrium is more likely to be reversed, the higher are the bene…ts of falling wages in the domestic production.
We conclude this section by illustrating the ambiguity on e in a nd ln e d ln t,
d ln w d ln t
o
space. Figure 5 illustrates Eqs. (39) and (40). For Eq. (39), the slope is clearly negative, whereas the slope of Eq. (40) depends on the sign of 00 ee( 0)
2
1 .
[Insert Figure 5 about here]
We take away from the graph that the more do domestic wages respond to changes in the o¤shoring costs, the more likely is a result contrary to the partial equilibrium case (an intersection of the two curves below the x-axis). By all means d ln e
d ln t < 0, for
00 ee( 0)
2 1 ,
which implies Eq. (40) to be horizontal or to be negatively sloped. If 00
> ee( 0)
2
1 , Figure 5
illustrates that the algebraic sign of d ln e
d ln t can be both negative or positive.
In the Appendix, we provide a numerical simulation of our model where we show that in fact, the result in partial equilibrium can be reversed in general equilibrium. Assuming speci…c parameter values and a linear cost function, we are able to document that there are cases in which d ln ed ln t < 0.
4
Conclusion
Although globalization of production has been discussed extensively in the literature, there is not yet a framework to study the relocation of whole varieties within the boundaries of a …rm. In this chapter, we show that the relocation of entire production lines leads to new insights into the labor market outcomes of o¤shoring. Reversing the assumptions that processes within a …rm are vertically related and that part of the production of a variety stays in the home country we have highlighted new multi-product speci…c transmission channels of o¤shoring. We set up a general oligopolistic equilibrium model of MPFs and o¤shoring, which allows us to study the consequences of globalization in the sense of declining costs of o¤shoring. We show that better prospects for o¤shoring a¤ect the geographic organization and the product range of an MPF. Giving a …rm the opportunity to o¤shore the production of labor-intensive products will lead to a broader product range. Considering the o¤shoring impacts on domestic employment, we highlight the cannibalization e¤ect of foreign on do-mestic output, which hits dodo-mestic employment next to the well established relocation e¤ect. Having wages endogenized, our model suggests ambiguous tendencies on the cuto¤ of pro-duction. The more do domestic wages respond to changes in o¤shoring costs and the higher are the bene…ts from lower wages in domestic production, the more likely is an even extended domestic production in an economy with increasing globalization. Therefore, our model is able to predict patterns in which …rms re-relocate entire product lines following globalization and a decline in o¤shoring costs.
One issue we did not consider in our model is welfare of workers. As our speci…cation considers domestic production only and consumption takes place on a third market, workers su¤er from declining wages and do not bene…t from lower prices of …nal goods. Due to this construction it does not make sense to assess welfare as we can not make any statements concerning the real wages in our model.
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5
Appendix
5.1
Proof of Lemma 1
In the open economy scenario, an MPF has to determine the pro…t maximizing geographic location of production. In the following, we will sketch this maximization problem. From the …rst-order condition for scale in Eq. (8), we know:
p(i) c(i) = b0
(1 e)x(i) + b0
eX, (A1)
which inserted in the open economy total pro…ts in Eq. (15) leads to
b0 = (1 e) e Z 0 x(i)2di + eX e Z 0 x(i)di + (1 e) Z e x (i)2di + eX Z e x (i)di. (A2) Given that X = e R 0 x(i)di +R e
x (i)di, we derive Eq. (17). To identify a condition for an optimally chosen cuto¤ variety e, we maximize Eq. (17) with respect to e. This implies the following …rst-order condition:
1 b0 d de = (1 e) 2 6 4 e Z 0 2x(i)dx(i) de di + Z e 2x (i)dx de di 3 7 5 (A3)
+(1 e)hx(e)2 x (e)2i+ (1 e)x ( )d
de + 2eX dX de = 0. With x ( ) = 0, dx(i) de = dx (i) de = e 1 e dX
de, and some mathematical conversion, we derive
1 b0
d
de = (1 e)
h
x(e)2 x (e)2i = 0. (A4)
5.2
Comparative Statics in Partial Equilibrium
In the following, we show how to derive the comparative static results of the model. In our model, the equilibrium is determined by the following system of equations:
x(i) = a 0 w (i) 2b0 eX 2b0 (1 e) (A6) x (i) = a 0 w ( (i) + t) 2b0 eX 2b0 (1 e) (A7) X = e Z 0 x(i)di + Z e x (i)di (A8) w ( ( ) + t) = a0 2b0 eX (A9)
We can reduce this system of equations to two equations in e and . In a …rst step, we substitute Eqs.(A6) and (A7) in Eq.(A8) and derive total output as
X = 1 2b0 1 8 > < > :a 0 w e Z 0 (i) di w 2 6 4 Z e (i) di + t e 3 7 5 9 > = > ;. (A10)
In a second step, we combine the latter expression with Eq. (A9) which leads to
w ( ( ) + t) = a0 e 1 8 > < > :a 0 w e Z 0 (i) di w 2 6 4 Z e (i) di + t e 3 7 5 9 > = > ;. (A11)
Eqs. (A5) and (A11) constitute two equations in two endogenous variables: e and . By totally di¤erentiating this system of equations, we derive our results in partial equilibrium. We show the total derivatives of Eqs. (A5) and (A11) in the next section of this Appendix.
5.3
Comparative Statics in General Equilibrium
In general equilibrium, we add the labor market clearing condition to our system of equations from the previous section. By substituting Eq. (A9) into the labor market clearing condition in Eq. (33), we derive
L = (w ( ( ) + t)) e
0
we 00
2b0(1 e) . (A12)
The combination of Eqs. (A5), (A11), and (A12) determines the general equilibrium of our model. In the total derivatives, we take into account that domestic wages are endoge-nously determined in the domestic labor market. For deriving the following results, note
that d de(e 0 ) = e and d de(e 00
) = e 2. Totally di¤erentiating the three equilibrium conditions Eqs. (A5), (A11), and (A12), with the results written as a matrix equation, we can analyze a change in the o¤shoring cost t as follows:
2 6 6 4 0 (w w ) 0 ~ ~ 1 0 e~ 0 ~ 0 w h ( ) ~ i ~ ~ 00 3 7 7 5 0 B @ 0( ) d ln td ln t ~d ln ~ w td ln t wd ln w w td ln t 1 C A = 0 B @ 1 2 ~ 0 1 C A . (A13)
The terms 1 and 2 are de…ned in Eq. (20) and are strictly positive. Using 2 = 00 02,
we can show that the determinant of the coe¢cient matrix is positive:
=nh(1 e) + e ~ i~ 00 + e~2 2o(w w ) 0 ~ + 1 ~ 2 w h ( ) ~ i > 0. (A14) In the following, we provide the solutions of the comparative statics exercise which we use in the general equilibrium part of our model.
E¤ect on Domestic Wages:
wd ln w w td ln t = 1 0 (w w ) 0 ~ 1 1 0 2 ~ 0 w h ( ) ~ i ~ ~ 0 (A15) wd ln w w td ln t = 1 n (w w ) 0 ~ e ~ ~ 0 + 1w h ( ) ~ i ~ o> 0 (A16)
E¤ect on Product Range:
0 ( ) d ln td ln t = 1 1 (w w ) 0 ~ ~ 2 0 e~ 0 ~ 0 w h ( ) ~ i ~ ~ 00 (A17) 0 ( ) d ln td ln t = 1 0 @ nh (1 e) + e ~ i~ 00 + e~2 2o(w w ) 0 ~ + 1 ~ e~ 0 w h ( ) ~ i ~ 1 A < 0 (A18)
E¤ect on Total Output: Totally di¤erentiating Eq. (A10) and using information from Eq. (A16) yields
2b0 1X w t d ln X d ln t = ~ ~ 0 n (w w ) 0 ~ e ~ ~ 0 + 1w h ( ) ~ i ~ o< 0. (A19) E¤ect on Cuto¤ Variety:
~d ln ~ w td ln t = 1 0 1 ~ 1 2 e~ 0 ~ 0 ~ 0 ~ 00 (A20)
Using again 2 = 00 02, we derive the following result:
d ln ~ w td ln t = 1 1 00 eh ~ ~ + ~ 0i 0 ? 0. (A21)
5.4
E¤ects of an Exogenous Change in Domestic Wages
This section keeps o¤shoring costs t constant and considers responses of the system of en-dogenously determined variables in Eqs. (9), (13), (14), (16), and (19) to changes in the domestic wage rate. Totally di¤erentiating this system of equations generates the following comparative statics results. It is apparent that with falling domestic wages total …rm output will increase, i.e.
d ln X d ln w = we 0 2b0 1X < 0. (A22)
This e¤ect gets larger the more domestic varieties bene…t from falling wages, i.e. the higher is 0
, and the more domestic varieties are produced onshore, i.e. the higher is e.
Changes in domestic factor prices clearly a¤ect the cuto¤ variety e as it is determined by the equality of production costs on- and o¤shore. With Home becoming a more attractive production site, more varieties will be manufactured domestically, i.e.
d ln e d ln w = w (w w ) (e) 0 e e < 0. (A23)
Akin to the previous result, we …nd that this e¤ect gets stronger the more the cuto¤ variety
varieties not being perfectly di¤erentiated (i.e. e > 0), foreign scale gets crowded out d ln x (i) d ln w = w 2b0(1 e) x (i) ee 0 1 > 0, (A24)
and the product range decreases as marginal varieties undergo cannibalization d ln
d ln w =
eew 0
1w 0( )
> 0. (A25)
The cannibalization e¤ect becomes stronger the more domestic production bene…ts from falling wages (i.e. the higher e and 0
). Figure 6 illustrates all e¤ects.
5.5
Numerical Example with a Linear Cost Function
In this section, we round down our analysis in general equilibrium with a numerical sim-ulation, where we focus on the ambiguity of the e¤ect of falling o¤shoring costs t on the cuto¤ variety ~. For speci…c parameter values and a linear cost function, Table 1 summa-rizes results for di¤erent degrees of product di¤erentiation. Results once again underline the issue of cannibalization in this framework. We observe a falling total …rm output X and a falling product range with rising substitutability between varieties (higher values of e). Referring to proposition 8 in the main body, it is important to mention that Table 1 shows a speci…c case where partial equilibrium results with respect to the cuto¤ variety e get reversed in general equilibrium, i.e. de
dt < 0. In this parameterization with an underlying
linear cost function, we …nd more varieties being produced onshore with falling o¤shoring costs. As explained before, this result is due to the prevailing e¤ect of falling domestic wages in comparison to the better prospects for o¤shoring.
Table 1: Numerical Example with a Linear Cost Function
Product di¤erentiation e w X e de dt d dt 0:1 8:70 121:73 1:36 22:13 0:688 0:495 0:5 8:14 36:13 1:77 8:86 1:081 0:003 0:9 9:01 22:96 1:14 2:91 0:370 0:143
Notes: Parameter values are: a0
= 100, b0
= 2, LW = 20, w = 3:5, and t = 2:5.
For this calculation, we assume a linear cost function: (i) = 0+ 1i = 1 + 0:5i.
Lemma 4 By assuming a linear cost function within this framework, we can show that there is the possibility that an MPF produces even more varieties domestically when it faces better prospects for o¤shoring.
i
δ
eX
b
a
'
−
2
'
)
(i
w
γ
0γ
w
Figure 1: Pro…t Maximizing Product Range
δ
eX
b
a
'
−
2
'
i
δ
eX
b
a
'
−
2
'
δ
~
i
c(i)
*c(i)
c(i)
a)
b)
c(i)
* oldδ
i
( )
i
x
*( )
i
x
δ
~
δ
Figure 3: Output Schedule
i
( )
i
x
*
( )
i
x
δ
~
δ
t
d
d
ln
~
ln
δ
t
d
w
d
ln
ln
Figure 5: Cuto¤ Variety in General Equilibrium
